Let be a stochastically continuous stationary max-stable process with Fréchet marginals . Here max-stable means that the finite dimensional distributions (fidi’s) of are max-stable multivariate distributions, see e.g., [3, 4, 5]. In the following is a non-negative process satisfying and has spectral process , i.e., we have the de Haan representation, see e.g., [3, 6] (below means equality in law)
where is a Poisson point process (PPP) on with intensity independent of ’s which are independent copies of . Since we consider here only stationary max-stable processes, adapting the terminology of  (motivated by ) we shall call the spectral process a Brown-Resnick stationary process.
The assumption that is stochastically continuous implies that it has a separable and measurable version, see e.g., ; the same holds for , see . Therefore in the following we suppose that both and are jointly measurable and separable. In the sequel we shall assume further that has locally bounded sample paths, and thus by (1.1) also has locally bounded sample paths. According to , this assumption is important for conditions that guarantee the existence of a dissipative Rosiński (or also called a mixed moving maxima) representation of .
By separability and the locally boundedness of the sample paths of and both and
are well-defined and finite random variables for any. Further, by (1.1) given we have (see e.g., [9, 10])
Since by measurability of , for any using Fubini theorem
with the Lebesque measure on , we have further that
The above shows that does not depend on the particular choice of the spectral process but only on . By the stationarity of it follows that
for any . Hence is sub-additive and thus by Fekete lemma
Moreover, from the above we conculde that does not depend on the particular choice of the spectral tail process but only on the stationary max-stable process . Referring to , is the so-called generalised Pickands constant defined with respect to a Brown-Resnick stationary process . In  is introduced for the log-normal process
is a centered Gaussian process with stationary increments, continuous sample paths and variance function. Taking to be a fractional Brownian motion (fBm) with self-similarity Hurst index , we get that is the classical Pickands constant, see e.g., [13, 14, 15, 11]. The only known values of are 1 and corresponding to and , respectively. In the case of Lévy processes the Pickands constant appears explicitly in [16, 17, 18, 11]. Moreover, for the discrete-time case we have that (introduced similarly as for the continuous-time, see ) is the extremal index of the stationary time series . In this context, it has been also studied in  using the spectral representation of
. A huge amount of research is dedicated to calculation and estimation of the extremal index of regularly varying time series, see e.g.,[21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and the references therein.
The main question that arises for Pickands constants , or in the discrete setup for the extremal index of a stationary max-stable time series is:
Q1: Under what conditions are these constants positive or equal to 0?
For the max-stable this question is partially answered in  for such that almost surely, and for being a non-negative random variable in . Specifically, the positivity of has been shown under the assumption that
In view of  since has locally bounded sample paths, then under (1.6) has a dissipative Rosiński representation which is equivalent with being generated by a non-singular dissipative flow, see [32, 33, 34, 6] for more details. As shown in [11, 31], if has càdlàg sample paths and (1.6) holds, then
with equality shown under some technical assumptions for both
Gaussian and Lévy case. The investigation therein was motivated by [35, 9].
The former contribution showed that
(1.7) holds with equality for an fBm.
Since in (1.4) is defined as a limit, it turns out that the explicit calculation of is for general too difficult. However, if (1.7) holds with equality, then being an expectation,
can be efficiently simulated. Indeed this has been successfully implemented for the classical Pickands constants in .
An interesting question that arises here is:
Q2: Does (1.7) hold with equality for general Brown-Resnick stationary ?
Clearly, if , then (1.4
) means the convergence in probability
whereas when we have the convergence in distribution
For being a symmetric -stable () stationary process with the above convergence has been shown in the seminal articles [1, 2], see the recent contributions [36, 37, 38, 39, 40, 41, 42] for related results and new developments.
The findings of [1, 2] are important for the max-stable processes too, which is already pointed out in  for discrete max-stable processes. Indeed, using the link between max-stable and processes established in  and  independently, it follows that when is generated by a non-singular conservative flow, which by  (under the assumption of locally boundedness of sample paths of ) is equivalent with then we have
Note that (1.10) holds also when we consider the discrete case , which can be shown for instance by utilising the expression of Pickands constant (which in this case coincides with the extremal index, [19, 31]) derived in .
We conclude that is positive if and only if . Moreover, as shown in  for the discrete case, can be explicitly given in terms of the spectral functions. Hence question Q1 has a simple answer, namely if is stationary max-stable process with locally bounded sample paths, then, by , if and only
Clearly, the convergence in probability in (1.8) implies that
as for any
and a similar implication holds for when (1.9) is satisfied. For being an random field the recent contribution  strengthened those convergences to that of for , i.e., showing the uniform integrability of whenever . The case that is a stationary max-stable random field is easier to deal with, see Proposition 3.13 in Section 4.
Our main interest in this contribution is the derivation of expressions for in terms of the spectral process that appears in the de Haan representation (1.1). In particular, motivated by Q2, we show that (1.7) (or a modification of it) holds with equality under (1.11) without further assumptions. Recall that so far it is only known that the inequality in (1.7) holds for having a dissipative Rosiński representation.
As already shown in [11, 31], different representations for relate to different dissipative Rosiński representations of . Therefore, our analysis is also concerned with such representations for .
Our study of Pickands constants (together with the criteria for its positivity) allows us to investigate the growth of the expectations of and as . The latter can be investigated under the further assumption of the Brown-Resnick model, i.e., when is a log-normal process. Moreover, for the Brown-Resnick model an extension of the celebrated Slepian inequality is possible, see Theorem 3.1 below.
Organisation of the paper: Our main results are displayed in Section 2 followed by discussions and some extensions displayed in Section 3. Proofs are postponed to Section 4; an Appendix concludes this contribution.
2. Main Results
Let be as in the Introduction being jointly measurable, separable and with locally bounded sample paths. By the measurability of we have that is a random variable in , see . Write next instead of for an event with . Fixing we have the following splitting formula
is Brown-Resnick stationary.
Since has also locally bounded sample paths and almost surely, the corresponding max-stable process is generated by a non-singular conservative flow. Moreover,
is also a Brown-Resnick stationary process which is generated by a
non-singular dissipative flow, provided that (1.11) holds. In order to omit technical details, we refer the reader to the deep contributions [34, 6] for details on conservative and dissipative parts of max-stable processes.
Consequently, by the discussions in the Introduction, condition (1.11) implies that
Therefore, in the following we can reduce our analysis by considering only the case that , i.e., is generated by a non-singular dissipative flow. In view of  this is equivalent with having a dissipative Rosiński representation i.e., for some non-negative random process (called also random shape function) which is continuous in probability (we can consider here therefore to be jointly measurable and separable) and for some we have
is a valid spectral process for , where is independent of and . See below Theorem 3.11 for a converse result. We state next the main result.
Let be a stochastically continuous max-stable process with de Haan representation (1.1). If has locally bounded sample paths with some spectral process satisfying condition (1.6), then there exists some jointly measurable and separable non-negative random shape function such that (2.4) holds and moreover we have
i) If is a stationary max-stable process, then the Pickands constant is defined by (see e.g., )
We shall show below that it is also possible to construct such that almost surely, and thus by (2.5) we obtain an alternative formula, namely
For simplicity we shall assume in the following that both and have càdlàg sample paths. Let be the space of càdlàg functions equipped with a metric which makes it complete and separable, see e.g.,  for details. Let be the Borel -algebra on defined by this metric and let be a probability measure given by
Since is a Polish metric space (complete and separable), we can determine a stochastic process with càdlàg sample paths and probability law ; refer to as the spectral tail process. This terminology is in agreement with  since as shown in , defined above agrees with the spectral tail process of the stationary time series , see [48, 49] for earlier and further results.
By [Thm 4.1] all the fidi’s of the max-stable process are determined by . Namely, we have the following inf-argmax formula valid for ’s positive constants and ’s in (see also Lemma 5.1 in Appendix for a short derivation)
Consequently, defines and vice-versa, from we can calculate the fidi’s of by the generalised Pareto distributions of , see [Remark 6.4] and  for representations of generalised Pareto distributions.
The next result gives an explicit construction for the random shape function and confirms (1.7).
Example 1. Consider the Gaussian case with as in (1.5), where is a centered Gaussian process with stationary increments, continuous sample paths and variance function .
In view of , the following condition
implies (1.1) and thus has a dissipative Rosiński representation with random shape function . Moreover
which has been proved in [Thm 2] under additional (redundant) assumptions on .
Example 2. Stationary max-stable Lévy–Brown–Resnick processes are introduced in , where the spectral process is constructed from two independent Lévy processes. Specifically, let be a Lévy process with Laplace exponent being finite for . Write for another independent Lévy process with Laplace exponent
Then we set , and if .
In view of  the max-stable process with unit Fréchet marginals corresponding to the spectral process is stationary.
Furthermore, by  the Lévy-Brown–Resnick process admits a dissipative Rosiński representation and thus Theorem 2.3 and [Thm 3.2] imply that (note that since almost surely then )
where is the bivariate Laplace exponent of the descending
ladder process corresponding to .
If is a spectrally negative Lévy process, by  , which agrees with the fact that
3. Discussions & Extensions
3.1. Slepian inequality for Brown-Resnick max-stable processes
Slepian inequality is essential in the theory of extremes and sample path properties of Gaussian and related processes. Besides it is also useful in numerous fields of mathematics including optimisation and number theory problems, see e.g., .
A commonly used version of Slepian inequality given for instance in [Thm 1.1] is as follows: If are two centered Gaussian processes, then for any we have
provided that for all
Moreover, in view of [Eq. 6] (applied to ) for any real-valued function
Max-stable processes that are constructed from log-normal Gaussian spectral processes are commonly referred to in the literature as Brown-Resnick max-stable processes, a first example is given in , see [4, 54, 5, 55, 56] for more details and interesting statistical models.
Both processes and are Brown-Resnick stationary, see [4, 32, 57, 58, 10] if defined by
where are two centered Gaussian processes with stationary increments and variance functions and , respectively. Since in general is different from we cannot use the refinement of Vitale  to Slepian inequality stated in (3.2) to arrive at (3.3).
Our next result states the Slepian inequality for Brown-Resnick stationary processes and . Moreover, it implies a comparison criteria for the corresponding Pickands constants.
It is well-known that the law of ’s depends only on their variograms , . Therefore we can suppose without loss of generality that for .
Let be two stationary max-stable Brown-Resnick processes with spectral processes and , respectively. Suppose that and are separable with locally bounded sample paths. If further for any
then for any compact set
i) As noted in , it is not known if there exists
any centered Gaussian process with stationary increments and variance function such that
ii) Under the setup of Theorem 3.1, if , then by Example 1, and consequently . This implies that cannot be generated by a non-singular conservative flow, therefore we can conclude that
iii) Theorem 3.1 extends the findings of [Thm 3.2].
iv) Slepian inequality has been extended to non-Gaussian setup in . Using the results of the aforementioned paper, Slepian inequality for max-stable processes can also be derived when and have fidi’s satisfying the conditions of [Thm 3.1].
3.2. Wills functional of Gaussian processes and asymptotic constants
In this section, motivated by Vitale , we discuss Wills functional for Gaussian processes
and its relation with Pickands and Piterbarg constants.
Consider as in the previous section, where is a centered separable Gaussian process with variance function and bounded sample paths. In  the Wills functional is defined as
This functional is important for derivation of lower bounds on supremum of Gaussian processes and random fields. Specifically, by [Thm 1] for any we have
where the last inequality follows by the boundedness of sample paths, see e.g., [(3.1)]. Our findings on Pickands constants, when is Brown-Resnick stationary can be utilised to derive asymptotic properties of Wills functionals as well as lower bounds for . Indeed, the Pickands constant is given in terms of Wills functional as
As shown in  is Brown-Resnick stationary if and only if has stationary increments. In view of Example 1, we have:
If is a centered separable Gaussian process with stationary increments, locally bounded sample paths and variance function , then
provided that (2.10) holds.
Sharp bounds on the expectation of maximum of Gaussian processes on finite intervals are obtained recently in ; see also . Our result above is of interest when considering the growth of supremum of on large increasing intervals and is proportional to for some (recall (2.10) which guarantees the positivity of ).
Wills functionals are tightly related with Borell-type bounds for tail distribution of suprema of Gaussian processes. We derive below a variant of Borell inequality accommodated to the setup of this section, see also [Cor 1]. For a given variance function , let in the following .
Suppose that is a centered separable Gaussian process with bounded sample paths and variance . For any such that and with we have
for any .
In the special case of having stationary increments, utilizing properties of Pickands constants, we arrive at the following corollary to Proposition 3.5. Set below , , with a standard fractional Brownian motion with Hurst index .
Suppose that is a centered Gaussian process with stationary increments, bounded sample paths and variance function . If for , with and , then for each
Note that is finite and asymptotically linear in as , by (3.7).
To this end we briefly discuss Piterbarg constants, which appear in the tail asymptotics of supremum of non-stationary Gaussian processes, see e.g., [63, 64].
Given as above, the corresponding Piterbarg constants are defined by
for some measurable locally bounded function and or . So far Piterbarg constants with have been reported and their finiteness for general under some restrictions on is recently shown in [Lem 4.5] and [Prop 3.1]. Clearly, the main question that arises is if is finite. For , its finiteness is shown using Piterbarg’s approach, see . We establish below the finiteness of Piterbarg constants for some general by using the fact that is Brown-Resnick stationary, and moreover derive a bound in terms of Wills functional. In the following proposition we consider only the case ; scenario follows by analogous line of reasoning.
If is as in Proposition 3.6, then for any measurable function such that and , we have that and moreover
3.3. Dissipative and conservative stationary max-stable processes
The decomposition of max-stable processes into conservative and dissipative parts has been recently discussed in  under some assumptions on the spectral process ; see also the recent review  for processes. Previous results based on the works of Rosiński [68, 69] have investigated decompositions of sum-stable and max-stable processes; see also [43, 34, 32]. Since we consider in this contribution the de Haan representation (1.1), we shall rely on the recent findings of . In [Lem 16] it is shown that a conservative/dissipative decomposition of corresponds to the so-called cone decomposition. In view of  a simple criteria for a stationary max-stable process with locally bounded sample paths to be generated by a non-singular conservative flow (abbreviate this as ” is conservative”) is , where is some spectral process of . The next two results propose conditions for as above to be conservative, or for to have a dissipative Rosiński representation. For simplicity, we consider the case that has càdlàg sample paths.
Let be stationary and max-stable process with Fréchet marginals and càdlàg sample paths. Denote by its spectral tail process. Then the following are equivalent:
The Pickands constant equals 0.
We state next equivalent conditions for to have a dissipative Rosiński representation.
3.4. Rosiński and De Haan Representations
In view of  and the above discussions, simple conditions on or guarantee the dissipative Rosiński representation (2.3). When has such a representation, using either  or our results here, we can construct the random shape function explicitly. Then, applying for instance [Thm 4.2] we can determine spectral processes as in (2.4) by choosing some random variable with positive density being independent of . Note that in the aforementioned reference with continuous sample paths are considered. The same holds under a more general assumption that is measurable.
If is a measurable non-negative process with locally bounded sample paths such that for some , then the max-stable processes corresponding to the spectral processes determined for different independent of via (2.4) are stationary with Fréchet marginals, equal in distribution, and have dissipative Rosiński representation (2.3).
The constant in the representation depends only on . Indeed, since we assume that has Fréchet distribution, by (1.2)
To this end we mention an important class of max-stable stationary processes for which is deterministic.
3.5. Growth of supremum
Below let be a separable max-stable stationary process with locally bounded sample paths and spectral process such that (1.1) holds. For any and we have for any
where stands for the Euler Gamma function and we used (1.3). Consequently, we have
If where or is a max-stable stationary process as above, then (3.13) holds and moreover is uniformly integrable for any .
3.6. Max-stable random fields
Max-stable random fields can be defined exactly as the max-stable processes by simply substituting the random process by a random field . The corresponding Pickands constant is then defined by
where again we suppose that is stationary max-stable with Fréchet marginals . Since the functional is translation invariant, i.e.,
and is sub-additive in the sense that for disjoint , then by 
we have that
exists and is finite.
In view of , with locally bounded sample paths has a dissipative Rosiński representation if , where with the Lebesgue measure in . Our findings above can be easily extended to this setup of stationary max-stable random fields. Note in passing that for the discrete-time setup the extremal index of is calculated in terms of spectral functions in , see also  for formulas in terms of spectral tail process.
Given the importance of the Gaussian model, for simplicity we state next the result for the Brown-Resnick max-stable model.
Let be a log-normal Gaussian random field where is a centered jointly measurable and separable Gaussian random field with stationary increments and variance function . If , which in light of  follows if
where stands for the Euclidean norm, then we have
Next, we discuss an important issue relevant for the simulation of Brown-Resnick max-stable stationary random fields, a topic of great interest for various applications, see for recent developments [72, 73, 9, 49, 74, 75]. As formulated in [Problem 2] for simulation of the Brown-Resnick max-stable stationary random field with spectral random field with a compact subset of , it is important to be able to minimise the functional
for all centered Gaussian random fields with stationary
increments and variance function that have the same variogram for some given continuous function .
We conclude this section with the following comparison result:
For two given separable centered Gaussian random fields with stationary increments, bounded sample paths and variance functions respectively, with a compact subset of , and for any measurable and locally bounded function , we have
provided that .
Proof of Theorem 2.1 We adapt the arguments of the proof of [Thm 2.1] for our max-stable process. Note first that by (1.11), in view of  has a dissipative Rosiński representation with some process which by the construction in the aforementioned paper is stochastically continuous and locally bounded (these properties are inherited from ). Therefore a jointly measurable and separable version of which is locally bounded exists, and we shall consider this version below.
Step 1: Since is given by (2.4), and moreover is locally bounded. then by (1.4) for any we have that